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We consider the problem of finding and describing minimisers of the Rayleigh quotient Lambda(infinity) := inf(u is an element of W2,infinity(Omega)\{0}) parallel to Delta(u)parallel to L-infinity(Omega)/parallel to u parallel to L-infinity(Omega), where Omega subset of R-n is a bounded C-1,C-1 domain and W-2,W-infinity(O) is a class of weakly twice differentiable functions satisfying either u = 0 on partial derivative Omega or u = parallel to Du parallel to = 0 on partial derivative Omega. Our first main result, obtained through approximation by L-p-problems as p -u003e infinity, is the existence of a minimiser u(infinity) is an element of W-2,W-infinity(Omega) satisfying {Delta u(infinity) is an element of Lambda(infinity)Sgn(f(infinity)) a.e. in Omega, Delta f(infinity) = mu infinity in Du0027 (Omega), for some f(infinity). L-1 (Omega) boolean AND BVl(oc)(Omega) and a measure mu(infinity) is an element of M(Omega), for either choice of boundary conditions. Here Sgn is the multi- valued sign function. We also study the dependence of the eigenvalue Lambda(infinity) on the domain, establishing the validity of a Faber- Krahn type inequality: among all C-1,C-1 domains with fixed measure, the ball is a strict minimiser of Omega -u003e Lambda(infinity)(Omega). This result is shown to hold true for either choice of boundary conditions and in every dimension.

The eigenvalue problem for the $$\infty $$-Bilaplacian